Re: possible problem with Lehmer pairs for the Riemann zeta function values

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Oct 24, 2025 at 12:34:13PM -0700, American Citizen wrote:
> Bill:
> 
> You made an astute observation on the rounding in base 2 versus base 10
> 
> The site says
> 
> The database of Riemann zeta zeros was computed by David Platt using the
> algorithm described in [MR:3315519, 10.1090/S0025-5718-2014-02884-6]. The
> imaginary part of each zero is  stored with an absolute precision of
> $±2^{−102}$ and the completeness of the list was verified using a rigorous
> version of Turing's method.

Thanks for the reference!

> But here we have 2^(-129) accuracy, at least 2^(-128) so this must be a typo
> ???
> 
> Guess I need to talk to David Platt and see if that 2^(-102) varies to
> 2^(-129) ??

Beware, it says 'absolute precision', while I used 'relative precision'.

So Platt claims that the zero imaginary part t satisfies
|t-388858886.0022851217767970582610330824021| <= 2^{-102}
Using t = 388858886.00228512177679705826103308240191063541672620941057947257917735912535
we find
? 388858886.00228512177679705826103308240191063541672620941057947257917735912535-388858886.0022851217767970582610330824021
%25 = -1.8936458327379058944E-31
? 2.^-102
%26 = 1.9721522630525295135293214132069655742E-31

so the absolute error is indeed < 2^-102
The relative error is 
|t/388858886.0022851217767970582610330824021-1| <= 2^-102/388858886.0022851217767970582610330824021
? abs(388858886.00228512177679705826103308240191063541672620941057947257917735912535/388858886.0022851217767970582610330824021-1)
%29 = 4.869750701098233153E-40
? 2.^-130
%31 = 7.3468396926392969248046033576390354864E-40
so is < 2^-130.
? 2^-102/388858886.0022851217767970582610330824021
%33 = 5.0716399548625467147596338022200743340E-40

In any case, Platt do not actually garanty that the two last digits are accurate,
as long as the absolute error is <=2^-102.

Cheers,
Bill